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Q13.
An investment account pays a fixed rate of 3% interest.
(a) How much would a £1000 investment be worth after 2 years?
(b) How long would it take for any amount invested to have doubled its value?
(a) - answer and commentary
3% interest is most easily written as 0.03. If we want to work out 3 percent of something we can multiply by 0.03. If we want to know
how much something is worth after a 3% increase we can multiply by 1.03 (that's the whole amount: 1 and the three percent
increase, 0.03).
At the end of the first year the £1000 will be worth £1000 x 1.03 = £1030
At the end of the second year you will have £1030 x 1.03 = £1060.90
We could have done all of that in one go: £1000 x 1.03 x 1.03 = £1060.90
Even neater than that: £1000 x 1.032 = £1060.90
Answer: £1060.90
(b) - answer and commentary
You can see from the answer to part (a) that we multiply by 1.032 after two years, so we must multiply by 1.033 after 3 years and so on.
So we need to know what power to raise 1.03 to in order to get a doubling, i.e. 2 or greater. That power will be the number of years.
Trial and improvement is fine:
1.033 = 1.09 - that's not remotely enough, so guess big!
1.0310 = 1.34 - still not enough!
1.0320 = 1.81 - getting there
1.0324 = 2.03 - a bit too much
1.0323 = 1.97 - too little
So, the first time it goes over 2 is after 24 years. Any investment will have doubled after 24 years at 3% interest.
For example, if you take our original investment of £1000, then we have:
£1000 x 1.0324 = £2032.79.
Answer: 24 years
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